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Section 1-8 : Logarithm Functions

In this section we’ll take a look at a function that is related to the exponential functions we looked at in the last section. We will look at logarithms in this section. Logarithms are one of the functions that students fear the most. The main reason for this seems to be that they simply have never really had to work with them. Once they start working with them, students come to realize that they aren’t as bad as they first thought.

We’ll start with \(b > 0\), \(b \ne 1\) just as we did in the last section. Then we have

When using property 7 above make sure that the logarithm that you subtract is the one that contains the denominator as its argument. Also, note that that we’ll be converting the root to fractional exponents in the first step.

You can use Property 8 on the second term because the WHOLE term was raised to the 3, but in the first logarithm, only the individual terms were squared and not the term as a whole so the 2’s must stay where they are!

The last topic that we need to look at in this section is the change of base formula for logarithms. The change of base formula is,

\[{\log _b}x = \frac{{{{\log }_a}x}}{{{{\log }_a}b}}\]

This is the most general change of base formula and will convert from base \(b\) to base \(a\). However, the usual reason for using the change of base formula is to compute the value of a logarithm that is in a base that you can’t easily deal with. Using the change of base formula means that you can write the logarithm in terms of a logarithm that you can deal with. The two most common change of base formulas are

In fact, often you will see one or the other listed as THE change of base formula!

In the first part of this section we computed the value of a few logarithms, but we could do these without the change of base formula because all the arguments could be written in terms of the base to a power. For instance,

This is a lot of work however, and is probably not the best way to deal with this.

So, in this section we saw how logarithms work and took a look at some of the properties of logarithms. We will run into logarithms on occasion so make sure that you can deal with them when we do run into them.